References & Citations

Bookmark

Mathematical Physics

Abstract: Pluri-Lagrangian systems are variational systems with the multi-dimensional
consistency property. This notion has its roots in the theory of pluriharmonic
functions, in the Z-invariant models of statistical mechanics, in the theory of
variational symmetries going back to Noether and in the theory of discrete
integrable systems. A $d$-dimensional pluri-Lagrangian problem can be described
as follows: given a $d$-form $L$ on an $m$-dimensional space, $m > d$, whose
coefficients depend on a function $u$ of $m$ independent variables (called
field), find those fields $u$ which deliver critical points to the action
functionals $S_\Sigma=\int_\Sigma L$ for any $d$-dimensional manifold $\Sigma$
in the $m$-dimensional space. We investigate discrete 2-dimensional linear
pluri-Lagrangian systems, i.e. those with quadratic Lagrangians $L$. The action
is a discrete analogue of the Dirichlet energy, and solutions are called
discrete pluriharmonic functions. We classify linear pluri-Lagrangian systems
with Lagrangians depending on diagonals. They are described by generalizations
of the star-triangle map. Examples of more general quadratic Lagrangians are
also considered.